Singularity of maps of several variables and a problem of Mycielski concerning prevalent homeomorphisms

Abstract

S. Banach pointed out that the graph of the generic (in the sense of Baire category) element of Homeo([0,1]) has length 2. J. Mycielski asked if the measure theoretic dual holds, i.e., if the graph of all but Haar null many (in the sense of Christensen) elements of Homeo([0,1]) have length 2. We answer this question in the affirmative.We call f∈Homeo([0,1]d) singular if it takes a suitable set of full measure to a nullset, and strongly singular if it is almost everywhere differentiable with singular derivative matrix. Since the graph of f∈Homeo([0,1]) has length 2 iff f is singular iff f is strongly singular, the following results are the higher dimensional analogues of Banach's observation and our solution to Mycielski's problem.We show that for d≥2 the graph of the generic element of Homeo([0,1]d) has infinite d-dimensional Hausdorff measure, contrasting the above result of Banach. The measure theoretic dual remains open, but we show that the set of elements of Homeo([0,1]d) with infinite d-dimensional Hausdorff measure is not Haar null. We show that for d≥2 the generic element of Homeo([0,1]d) is singular but not strongly singular. We also show that for d≥2 almost every element of Homeo([0,1]d) is singular, but the set of strongly singular elements form a so called Haar ambivalent set (neither Haar null, nor co-Haar null).Finally, in order to clarify the situation, we investigate the various possible definitions of singularity for maps of several variables, and explore the connections between them.